Spiegelhater et al. [1]} systematically investigate the model complexity of Bayesian hierarchical models. A unique challenge in measuring the complexity of Bayesian hierarchical models is that the number of model parameters is not clearly defined. Spiegelhater et al. [1]} define a model complexity measure by the number of effective parameters. Using the information theoretic argument, they show that this complexity measure can be estimated by the difference between the posterior mean of the deviance and the deviance at the posterior estimates of the parameters of interest, and is approximately the trace of the product of Fisher's information [3]} and the posterior covariance matrix. In addition, they suggest that adding such a complexity measure to the posterior mean deviance gives a deviance information criterion for comparing models.
